Let $\displaystyle (a_{n} ; n\ge 1)$,

$\displaystyle (b_{n}; n\ge 1)$ and

$\displaystyle (l_{n}; n\ge 1)$

be three sequences of real numbers. Supose that

$\displaystyle l_{n} \in [0,1]$

for every n, and let

$\displaystyle c_{n}=l_{n}a_{n}+(1-l_{n})b_{n}$.

Assuming that

$\displaystyle lim sup a_{n}$ and

$\displaystyle lim sup b_{n}$ are finite, please

prove the following inequality:

$\displaystyle lim sup c_{n} \le max(lim sup a_{n},lim sup b_{n})$